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This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by M. Artin and B. Mazur. Eric M. Friedlander presents many of his own applications of this theory to algebraic topology, finite Chevalley groups, and algebraic geometry. Of particular interest are the discussions concerning the Adams Conjecture, K-theories of finite fields, and Poincare duality. Because these applications have required repeated modifications of the original formulation of etale homotopy theory, the author provides a new treatment of the foundations which is more general and more precise than previous versions.One purpose of this book is to offer the basic techniques and results of etale homotopy theory to topologists and algebraic geometers who may then apply the theory in their own work. With a view to such future applications, the author has introduced a number of new constructions (function complexes, relative homology and cohomology, generalized cohomology) which have immediately proved applicable to algebraic K-theory.

Algebraic topology --- 512.73 --- 515.14 --- Homology theory --- Homotopy theory --- Schemes (Algebraic geometry) --- Geometry, Algebraic --- Deformations, Continuous --- Topology --- Cohomology theory --- Contrahomology theory --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Homotopy theory. --- Homology theory. --- Abelian group. --- Adams operation. --- Adjoint functors. --- Alexander Grothendieck. --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number theory. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Automorphism. --- Base change. --- Cap product. --- Cartesian product. --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Comparison theorem. --- Complex number. --- Complex vector bundle. --- Connected component (graph theory). --- Connected space. --- Coprime integers. --- Corollary. --- Covering space. --- Derived functor. --- Dimension (vector space). --- Disjoint union. --- Embedding. --- Existence theorem. --- Ext functor. --- Exterior algebra. --- Fiber bundle. --- Fibration. --- Finite field. --- Finite group. --- Free group. --- Functor. --- Fundamental group. --- Galois cohomology. --- Galois extension. --- Geometry. --- Grothendieck topology. --- Homogeneous space. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Inverse system. --- K-theory. --- Leray spectral sequence. --- Lie group. --- Local ring. --- Mapping cylinder. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Newton polynomial. --- Noetherian ring. --- Open set. --- Opposite category. --- Pointed set. --- Presheaf (category theory). --- Reductive group. --- Regular local ring. --- Relative homology. --- Residue field. --- Riemann surface. --- Root of unity. --- Serre spectral sequence. --- Shape theory (mathematics). --- Sheaf (mathematics). --- Sheaf cohomology. --- Sheaf of spectra. --- Simplex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Surjective function. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Tubular neighborhood. --- Vector bundle. --- Weak equivalence (homotopy theory). --- Weil conjectures. --- Weyl group. --- Witt vector. --- Zariski topology. --- Homologie --- Topologie algebrique --- Geometrie algebrique --- Homotopie